Artin approximation compatible with a change of variables

TL;DR

This paper introduces a modified Artin approximation that allows perturbing variables in Nash equations, enabling Nash solutions to approximate formal solutions compatibly with variable changes, with applications in Nash map equivalence.

Contribution

It presents a new version of Artin approximation compatible with variable changes in Nash equations, extending nested approximation concepts to real and complex geometry.

Findings

Nash solutions can approximate formal solutions with variable perturbations.

The approximation is valid in both local and global real and complex settings.

Application to Nash map equivalence demonstrates practical utility.

Abstract

We propose a version of the classical Artin approximation which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a Nash equation by a Nash solution in a compatible way with a given Nash change of variables. This results is closely related to the so-called nested Artin approximation and becomes false in the analytic setting. We provide local and global version of this approximation in real and complex geometry together with an application to the Right-Left equivalence of Nash maps.